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Theorem elprn1 40183
Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elprn1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)

Proof of Theorem elprn1
StepHypRef Expression
1 neneq 2829 . . 3 (𝐴𝐵 → ¬ 𝐴 = 𝐵)
21adantl 481 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → ¬ 𝐴 = 𝐵)
3 elpri 4230 . . . 4 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
43adantr 480 . . 3 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (𝐴 = 𝐵𝐴 = 𝐶))
54ord 391 . 2 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → (¬ 𝐴 = 𝐵𝐴 = 𝐶))
62, 5mpd 15 1 ((𝐴 ∈ {𝐵, 𝐶} ∧ 𝐴𝐵) → 𝐴 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  fourierdlem70  40711  fourierdlem71  40712  fouriersw  40766  prsal  40856  sge0pr  40929
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